On Categorical Models of Classical Logic and the Geometry of Interaction

Transcription

1 On Categorical Models o Classical Logic and the Geometry o Interaction Carsten Führmann David Pym January 18, 2005 bstract It is well-known that weakening and contraction cause naïve categorical models o the classical sequent calculus to collapse to oolean lattices In previous work, summarized briely herein, we have provided a class o models called classical categories which is sound and complete and avoids this collapse by interpreting cut-reduction by a poset-enrichment Examples o classical categories include boolean lattices and the category o sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product In this article, which is sel-contained, we present an improved axiomatization o classical categories, together with a deep exploration o their structural theory Observing that the collapse already happens in the absence o negation, we start with negation-ree models called Dummett categories Examples include, besides the classical categories above, the category o sets and relations, where both conjunction and disjunction are modelled by the disjoint union We prove that Dummett categories are MIX, and that the partial order can be derived rom hom-semilattices which have a straightorward proo-theoretic deinition Moreover, we show that the Geometry-o-Interaction construction can be extended rom multiplicative linear logic to classical logic, by applying it to obtain a classical category rom a Dummett category long the way, we gain detailed insights into the changes that proos undergo during cut elimination in the presence o weakening and contraction Contents 1 Introduction 2 11 Outline 5 12 Related work 6 2 Preliminaries 7 21 The sequent calculus 7 22 Proo nets Net-equivalence Symmetric linearly distributive categories Categorical semantics o MLL The interpretation o sequents Nets as symmetric linearly distributive categories 20 1

2 3 Modelling weakening and contraction: Dummett categories Symmetric monoids and comonoids MIX Poset-enrichment Homomorphisms The structure o Dummett categories Duality o the monoids and comonoids Compact Dummett categories Nets or symmetric monoidal categories Characterizing compact Dummett categories by one equality Dummett categories with inite biproducts 50 4 Geometry o interaction in the presence o weakening and contraction Traced symmetric MIX categories The traditional GoI construction The GoI construction extended to traced Dummett categories GoI or traced categories with inite biproducts 61 5 Directions or uture work 64 Some lemmas and proos 69 1 Introduction It is notoriously hard to ind a decent denotational semantics or the classical sequent calculus, let alone an algorithmic interpretation This problem is related to the non-deterministic behaviour o cut elimination To see the point, consider the ollowing sequent proo: Λ = Φ 1 Γ weakening Γ, Γ, Γ, contractions Γ Φ 2 Γ weakening Γ, Cut where Φ 1 and Φ 2 are arbitrary proos o the sequent Γ We call this the Laont proo, because it is a variant o an example credited to Laont (c (Girard, Laont & Taylor 1989, p 151)) The sub-proo Φ 1 is weakened on the right, and the sub-proo Φ 2 is weakened on the let Then ollows a cut, where the cut ormula is the ormula introduced by the weakenings Finally, the double occurrences o Γ and are removed by let and right contractions (Clearly, the two contractions are supposed to commute with each other, so we need not be speciic about the order in which they are applied) The proo Λ, 2

3 reduces to Φ 1 Γ weakenings Γ, Γ, or to Φ 2 Γ weakenings Γ, Γ, contractions contractions Γ Γ ut, clearly, the weakenings ollowed by the contractions are essentially nothing (c (Girard et al 1989, p 152)) So Φ 1 and Φ 2 are obtained by reducing the same proo Thus, the denotations o Φ 1 and Φ 2 must be equal or any semantics that admits cut-reduction in the sense that the reduct is denotationally equal to the redex In summary, any denotational semantics that admits cut-reduction must identiy all proos o a sequent Γ Note that this argument does not rely on negation! There are various escapes rom this denotational collapse: irst, we might simply abandon classical logic and adopt, or example, intuitionistic logic or linear logic instead s explained in Gentzen s seminal article (Gentzen 1934), intuitionistic logic can an be obtained by restricting the classical sequent calculus is such a way that the succedent contains at most one ormula s is widely known, intuitionistic logic can be modelled by cartesian-closed categories Models o linear logic also abound ut both intuitionistic logic and linear logic dier rom classical logic with respect to provable sequents, and we do not wish to depart rom classical provability Second, insisting to keep classical logic, we might move to classical natural deduction systems (Prawitz 1965), where proos may be represented as terms o the λµν-calculus (Parigot 1992, Pym & Ritter 2001) ut such systems do not admit all cut-reductions: as it turns out, the call-by-name version o λµν admits only the reduction to Φ 2, while the call-by-value version admits only the reduction to Φ 1 Each version corresponds to a dierent choice o -translations (aka continuation-passing-style transorms in programming-language jargon) o classical logic into intuitionistic logic (Troelstra & Schwichtenberg 1996, Plotkin 1975) Models o λµν can be obtained in ibrations over a base category o structural maps in which each ibre is a model o intuitionistic natural deduction and in which dualizing negation is interpreted as certain maps between the ibres (Ong 1996, Pym & Ritter 2001) lternative models are given by control categories and co-control categories (Selinger 2001) In our companion paper (Führmann & Pym 2004b), we presented a solution that, unlike classical natural deduction, models all cut-reductions: we introduced a kind o poset-enriched category called classical category whose objects model types and whose morphisms model proos o the classical sequent calculus; whenever a proo o Φ can be reduced to another proo Ψ, we only require C Φ C Ψ (as opposed to C Φ = C Ψ ), where C Φ and C Ψ are the morphisms denoted by Φ and Ψ in the classical category C Classical categories are a special case o symmetric linearly distributive categories (Cockett & Seely 1997b): they have symmetric monoidal products and or modelling conjunction and disjunction, respectively To model contraction and 3

4 weakening on the right, every object is endowed with a symmetric monoid ( :, [] : ); the multiplication models contraction, and the unit [] models weakening Dually or contraction and weakening on the let (It is worth mentioning here that symmetric linearly distributive categories with negation are equivalent to -autonomous categories; however, the ormer provide better choice o primitives or achieving our goals) In (Führmann & Pym 2004b), we proved that classical categories are sound and complete or the classical sequent calculus More precisely, we introduced a notion o theory with judgments o the orm Φ Γ Ψ, Γ where the is a preorder that contains all reductions required or cut elimination The soundness theorem in (Führmann & Pym 2004b) states essentially that Φ Ψ implies C Φ C Ψ or every classical category C The completeness theorem states essentially that Φ Φ is a theorem o a theory T whenever C Φ C Ψ holds or every model C o T Its proo uses a category built rom Robinson s proo nets or classical logic (Robinson 2003), which correspond directly to the classical sequent calculus (We shall discuss these nets briely in 22) morphism o that category is an equivalence class o proo nets with respect to the preorder For morphisms, g : with representing nets N and N g, that category has g i and only i N N g (This explains why is a partial order although the preorder is not generally antisymmetric) In (Führmann & Pym 2004b), we gave the ollowing concrete examples o classical categories: n initial model built rom proo nets; The category Rel o sets and relation, where both and are deined to be the evident unctor that takes two sets to their cartesian product, and is the set-theoretic inclusion o relations; oolean lattices The product o any two classical categories or example, Rel or any oolean lattice This shows that there are models which are non-posetal (ie, there are hom-sets with more than one element) and non-compact (ie, ) In (Führmann & Pym 2004a), we ound urther classical categories that arise rom an abstract Geometry-o-Interaction (GoI) construction starting with a quantaloid, and used those models to study the increase o denotations during cut elimination Since we presented (Führmann & Pym 2004a) in the summer o 2004, we managed to (1) greatly advanced the axiomatization and understanding o classical categories, in particular by proving that they are MIX, and (2) strongly generalize the GoI construction we presented in (Führmann & Pym 2004a) Many o the new insights were sparked by Masahito Hasegawa in private communications, which is why several propositions in this article are attributed to him 4

5 This article gives a comprehensive account o our improved axiomatization and structural theory o classical categories ( 3), and o our generalized GoI construction ( 4) Owing to the substantial advances o presentation and axiomatization, we chose to make this article sel-contained and require no previous knowledge o (Führmann & Pym 2004b) or (Führmann & Pym 2004a) On the purely technical side, we have adopted the proo nets in the style o (lute, Cockett, Seely & Trimble 1996); understanding these nets takes a little more eort than understanding Robinson s nets, but they are more eicient or calculations 11 Outline Now or a detailed overview o this article In 2, we recall some preliminaries: the classical sequent calculus, proo nets, and the categorical semantics o multiplicative linear logic (MLL) in symmetric linearly distributive categories In 3, we introduce classical categories rom the ground up We proceed in two steps: irst, we extend symmetric linearly distributive categories presented with structure or modelling weakening and contraction This structure consists o a symmetric monoid and a symmetric comonoid or every object, and a poset-enrichment The resulting categories are models o the negation-ree ragment o the classical sequent calculus We call them Dummett categories (inspired by Dummett s extensive discussion, in Elements o Intuitionism (Dummett 1977), o multi-succedent intuitionistic sequent calculi) Second, we introduce classical categories as Dummett categories with the property o having negation in the sense o Cockett and Seely We then establish the close connection between classical categories and the classical sequent calculus by constructing the ree classical category rom proo nets (Theorem 332) (This extends the construction o the ree symmetric linearly distributive category rom MLL proo nets in the sense o (lute et al 1996)) From a logical point o view, the result means that classical categories are sound and complete (in order-theoretic sense explained above) with respect to a certain super-relation o cut-reduction or the classical sequent calculus Our ree construction relies on a series o results about the structure o Dummett categories, including the remarkable result (due to Hasegawa) that the monoids or comonoids cause symmetric linearly distributive categories to be MIX (Theorem 311); the act that the poset-enrichment is not needed as extra structure, but is induced by hom-semilattices which are derivable rom other primitives (Prop 328) We inish 3 by presenting a extremely economic axiomatization o compact Dummett categories (Prop 342, due to Hasegawa), and even more economical axiomatization o Dummett categories with inite biproducts (Prop 343, also due to Hasegawa) 5

6 In 4, we introduce an extended GoI construction that sends a traced Dummett category to a classical category (Theorem 44) This shows that GoI works in the presence o weakening and contraction, even with respect to the partial order that models cut-reduction s we shall explain, traced Dummett categories are essentially traced symmetric monoidal categories, plus symmetric monoids and symmetric comonoids on every object, satisying certain conditions Our extended GoI construction is an instance o the well-known construction o a compact closed category rom a traced symmetric monoidal category (For an overview o the history o GoI leading to that construction, see the introduction o 4) The key point o our extended construction is that the symmetric monoids and symmetric comonoids, and the conditions required or a Dummett category, survive the extended GoI construction In 44, we study the special case where the starting point o the extended GoI construction is a traced Dummett category with inite biproducts In particular, we present a comprehensive characterization o morphisms in such GoI categories with respect to their behaviour under cut-reduction (Prop 45) In 5, we suggest some directions or uture work 12 Related work The article (Hyland 2004) introduces a notion o abstract interpretation o classical proo as a compact closed category in which every object is equipped with a symmetric monoid and a symmetric comonoid, satisying certain conditions (This work was oreshadowed in (Hyland 2002)) These abstract interpretations are almost the same as our classical categories in the compact case where = The only dierence is that compact classical categories need to satisy an extra equation (Equation 3 in 342) s we shall show in 342, this equation implies that every compact classical category has hom-semilattices, which yield the partial order we use or modelling cut-reduction So our approach is more general than Hyland s in that it does not require compactness, and more special in that we require certain conditions that lead to the existence o hom-semilattices nother overlap with (Hyland 2004) happens where we specialize our GoI construction to categories with inite biproducts The partial order speciic to our models allows a precise analysis o the behaviour o morphisms with respect to cut-reduction (explained in 321) The article (ellin, Hyland, Robinson & Urban 2004) contains a semantics o the classical sequent calculus which is iner grained than ours in that it rejects axiom expansions (also called η-rules), that is, the categorical connectives and which model conjunction and disjunction do not generally preserve identities In contrast, our work its into the existing ramework o symmetric linearly distributive categories, in which and are unctorial nother dierence between our work and (ellin et al 2004) is that we deal with modelling cut-reduction (using the poset-enrichment) whereas (ellin et al 2004) do not In (Dosen 1999, Dosen & Petric 2004), a notion o oolean category is introduced This notion relies on the presence o products and coproducts, leading 6

7 to a more collapsed structure than ours, closely related to the category o inite sets and relations There is also some interesting work about conluent cut elimination in the presence o the MIX rule (ellin 2003, Lamarche & Straßburger 2004) For example, one can remove the non-determinism o cut-reduction by allowing a reduction Φ Γ WR Γ, Γ, Γ, Φ Γ WL Γ, Cut Φ Φ Γ Γ MIX Γ, Γ, The conluent cut elimination procedure in (Lamarche & Straßburger 2004) (which is based on proo nets) does this implicitly Our semantics is compatible with this approach: the MIX rule is denotationally equivalent to a degenerate cut with cut ormula = or = (oth choices o result in the same denotation) So, in our view, this kind o conluent cut elimination is a removal o arbitrary cuts in avour o degenerate cuts (ie, MIXes); a MIX is still non-deterministic in act, it is the pure incarnation o proo-theoretic non-determinism, because it is the parallel composition o Φ and Φ that one might want to reduce to either Φ or Φ Our models support this view, because they admit the reduction o MIX to Φ and to Φ In act, the hom-semilattices o our models are given by Φ 1 Φ 2 Γ Γ Φ 1 Φ 2 = MIX Γ, Γ, contractions Γ From a technical point o view, this article is based on symmetric linearly distributive categories, which were introduced in (Cockett & Seely 1997b) In particular, we heavily use the proo nets or symmetric linearly distributive categories introduced in (lute et al 1996), because they are very eicient or the calculations required in this article We also build on the discussions o MIX categories in (lute, Cockett & Seely 2000) and (Cockett & Seely 1997a), and the notion o traced object in a MIX category presented in (lute et al 2000) We also rely on results rom the GoI literature; the related work in this area is described in 4 2 Preliminaries 21 The sequent calculus The version o the sequent calculus to which we reer is given in Tables 1 and 2 We use the system o multiplicative linear logic (MLL) presented in Table 1, plus the rules or weakening and contraction presented in Table 2, and so obtain 7

8 x L Γ,,, Γ Γ,, Γ Γ 1 1, 1, 1 Γ 2 2, 2, 2 Γ 1, Γ 2 1, 2, 1 2, 1, 2 R L Γ 1, 1, Γ 1 1 Γ 2, 2, Γ 2 2 Γ 1, Γ 2, 1 2, Γ 1, Γ 2 1, 2 Γ,,, Γ,, R L Γ, Γ Γ,, Γ R L Γ, Γ,, R L Γ, Γ,, Γ,, Γ, R Γ,, Γ, Γ, Γ,, EL Γ,,, Γ Γ,,, Γ Γ,,, Γ,,, ER Γ 2 1,, 3 Γ 1,, Γ 3 2 Cut Γ 1, Γ 2, Γ 2 1, 2, 3 Table 1: Inerence rules o MLL WL Γ, Γ Γ,, Γ Γ, Γ,, WR CL Γ,,, Γ Γ,, Γ Γ,,, Γ,, CR Table 2: Inerence rules or weakening and contraction 8

9 a calculus which diers rom LK (Gentzen 1934) only in its use o the multiplicative orm o the introduction rules and the absence o implication We consider implication to be derived that is, = sequent has the orm Γ where Γ and are inite lists o ormulæ The capital Latin letters range over ormulæ Henceorth, we shall call sequent proos derivations, to avoid clashes with the notion o proo at the meta-level To acilitate semantics, we shall introduce a more economic version o the sequent calculus just described: the new version is obtained by replacing the rules R, L, L, R, WL, WR, CL, and CR by axioms For example, to replace R, we introduce an axiom and consider R as an abbreviation or x R, Γ 2 2, 2, 2 1, Cut x R Γ 1 1, 1, 1 1, Γ 2 2, 1 2, 2 Cut Γ 1, Γ 2 1, 2, 1 2, 1, 2 The extra axioms lead to the revised version o the sequent calculus described in Tables 3, 4, and 5 (We put the rules or negation in a separate table because we shall also study the negation-ree ragment) This revised version acilitates semantics, because axioms simply denote morphisms, and only seven inerence rules remain which are not axioms However, we shall keep the names R, L, L, R, WL, WR, CL, and CR as abbreviations or the evident derivations that involve x R, x L, x L, x R, xwl, xwr, and xcl, respectively For the purpose o categorical logic, we shall consider derivations over any signature signature Σ consists o a set o atomic ormulæ and a set o optional axioms The set o ormulæ over Σ is generated in the evident way rom the atomic ormulæ, using,,,, and We call a ormula over Σ positive i it is negation-ree Optional axioms are o the orm Γ Typical optional axioms are the ones or weakening and contraction in Table 5 Deinition 21 derivation Φ over a signature Σ is a tree generated by the rules in Tables 3 and 4, plus the optional axioms o Σ We call a derivation over Σ positive i all o its ormulæ are positive 22 Proo nets The essence o a sequent proo can be captured by a proo net, an idea introduced by Girard (Girard 1987) (or net or short) In this article, we shall need proo nets to describe equalities between proos The nets we use are, essentially, those rom (lute et al 1996), extended to account or the additional structural properties o classical logic Thus in this paper we depart rom our use, in (Führmann & Pym 2004b), o the classical proo nets introduced by 9

10 x L Γ,,, Γ Γ,, Γ, x R x L, Γ,,, Γ,, R L Γ, Γ Γ,, Γ R L Γ, Γ,, R EL Γ,,, Γ Γ,,, Γ Γ,,, Γ,,, ER Γ 2 1,, 3 Γ 1,, Γ 3 2 Cut Γ 1, Γ 2, Γ 2 1, 2, 3 Table 3: Revised inerence rules o MLL: negation-ree ragment x L,, x R Table 4: Revised inerence rules o MLL: axioms or negation 10

11 xwl xwr xcl xcr Table 5: Optional axioms or weakening and contraction Robinson (Robinson 2003) Robinson s nets correspond more directly to the sequent calculus than those o (lute et al 1996), the latter being more convenient or calculations Inormally, a net is a graphical skeleton o a derivation For example, both derivations x C, C R x L x R, x C C L, ( C) ( ), C L ( C) ( ), C R ( C) ( ) C have the ollowing proo net: and, ( C) ( ), C R, ( C) ( ) C L ( C) ( ) C x x x : L : L C : L : R : R C : R L C : L R : R L R ( C) : L ( ) C : R This net is in the style used in (Robinson 2003); in that paper, a proo structure is deined to be a bipartite directional graph whose two amilies o nodes are labelled as ollows: Family 1 labelled by an inerence rule o the sequent calculus; Family 2 labelled by a ormula, together with the inormation Let o Right The graph is subject to two additional constraints, which essentially mean that 1 the incoming (resp outgoing) arrows o a rule node uniquely match the hypothesis (resp conclusions) o the corresponding rule o the sequent calculus; 11

12 2 each ormula node has a unique incoming and at most one outgoing arrow Translating derivations into proo structures is straightorward Not all proo structures are the images o derivations; those that are called proo nets (When a graph is a proo net can also be characterized by the switching criterion introduced in (Danos & Regnier 1989), which requires that certain subgraphs o the proo structure be connected and acyclic) Robinson s nets, with minor notational changes, were used in (Führmann & Pym 2004b) However, in this article, we have adopted the nets introduced in (lute et al 1996) In that style, the net or the derivations above ( C) C C ( ) C Here, the only nodes are rule nodes We write or, or, and or, because these nets are also used to describe morphisms in symmetric linearly distributive categories, as we shall see in 242 The wires are labelled with types, which can be seen either as ormulæ or as objects o a symmetric linearly distributive category The let-hand ormula o the derived sequent appears at the top o the net, and the right-hand ormula at the bottom The top-to-bottom orientation has advantages over the let-to-right orientation with respect to the alignment o types and wires It also ensures the nice property that a net is planar i and only i the corresponding derivations are within non-commutative logic, that is, they contain no exchange rules, c (lute et al 1996) n important dierence between nets in the style o Robinson and nets in the style o (lute et al 1996) is that the latter have no axiom links and no cut links bandoning these links is possible because a cut and an axiom cancel each other out according to a (poly)categorical neutrality law (Führmann & Pym 2004b) nother dierence is that Robinson s nets have links or weakening and contraction, while nets in the style o (lute et al 1996) do not (However, we shall see that such links can be easily added to the latter) It is a bit harder to make the leap rom derivations to nets in the style o (lute et al 1996) than to nets in the style o Robinson However, the ormer are better or heavy calculations, because they have no cluttering cut links and axiom links, and because one can drop the type annotations when they are clear rom the context (Just as one sometimes omits type annotations o lambda terms) This is why we opted or them in this article Now we turn towards a ormal deinition o nets, based on the deinition in (lute et al 1996), but not quite as ormal We deine the notion o typed circuit uilding a typed circuit requires a set T o types and a set C o components Each component C has a list α = ( 1,, n ) o types describing the input 12

13 ports, and a list β = ( 1,, m ) o types describing the output ports o 1 1 n m We deine the collection o circuits over C inductively: Every component C is a circuit The identity wire is a circuit, with one input port and one output port, each o type Given any number o circuits, connecting some output ports with input ports o the same type yields another circuit For and example circuit, consider C D F H g E I J K G Note that it has two connections (o types F and H) rom to g s we shall see, this is not a net or symmetric linearly distributive categories, because those nets must have exactly one connection between any two components; however, the nets or symmetric monoidal categories that we shall introduce much later in 341 allow such multiple connections Remark 22 Our deinition o circuit is more general than that in (lute et al 1996) in that it allows eedback, eg, which we shall employ only in 4 13

14 net (or symmetric linearly distributive categories), in short, is a circuit built rom components that correspond to the introduction rules o the sequent calculus, subject to the condition o sequentiality, which means that the circuit must represent a derivation We shall now spell this out in detail The types or nets are given by the grammar, ::= b, where b ranges over atomic ormulæ We have components L R or conjunction, and components L R or disjunction We have components L R to deal with Remark 23 curiosity here is that L requires the supporting wire The wire that is directly attached to the supporting wire is called thinning link in (lute et al 1996) Thinning links are needed because o categorical coherence issues: or example, using nets without thinning links would orce the identity morphism on to be equal to the twist map, which is alse in some symmetric linearly distributive categories (see (lute et al 1996)) Dually, we have components L to deal with The components L and R are called thinning links When we consider negation, we also use components R L R Table 6 describes how a derivation Φ o 1,, n 1,, m is turned into a circuit 1 n Φ 1 m 14

15 The double lines labelled Γ i or j stand or bundles o wires, one or every ormula contained in Γ i or j In the translations or L and R, any wire in Φ can be used as a supporting wire (We shall consider any two choices o supporting wire to be equivalent, see 221 ) We call the components L, R, L, R, L, R, L, R, L, and R links to distinguish them rom arbitrary components Links depicted by rectangular boxes correspond to axioms (eg x R, x L, x L, x R, L, R); they are nets Links with circles correspond to inerence rules which have one or more hypotheses; they are used to build nets, but they are not nets (This is a notational clariication we adopt rom (Cockett, Koslowski & Seely 2003)) Deinition 24 net over a signature Σ is a circuit whose types are the ormulæ over Σ, whose components are the links L, R, L, R, L, R, L, R, L, R, and components o the orm 1 n, 1 m where 1,, n 1,, m is an optional axiom o Σ, which is in the image o translation in Table 6 We call a net positive i all o its ormulæ are positive We write Net(Σ) (resp Net (Σ)) or the positive (resp arbitrary) nets over Σ We write Net(Σ)(Γ, ) (resp Net (Σ)(Γ, )) or the positive (resp arbitrary) nets with input ports according to Γ and output ports according to 221 Net-equivalence In this section, we shall recall the equivalence between nets introduced in (lute et al 1996) It is deined by a number o rules or rewriting subcircuits These rules can only be applied i the both original circuit whose subcircuit is rewritten and the resulting circuit are nets First, we have reductions that simulate the cut elimination o MLL: Reduce = = Reduce Reduce = = Reduce 15

16 Φ Γ 2 1,, 3 Ψ Γ 1,, Γ 3 2 Cut Φ = Ψ x = Γ 1, Γ 2, Γ 2 1, 2, Γ 1 Γ 2 Γ 3 Φ Γ 1,,, Γ 2 L Γ 1,, Γ 2 = Γ 1 Γ2 x R, Φ = x L, = Φ Γ 1,,, 2 R = Γ Φ Γ 1,, Φ Γ 1, Γ 2 L Γ 1,, Γ 2 = Γ 1 Φ Γ2 R = L = Φ Γ 1, 2 R Γ 1,, 2 = Φ 1 2 Γ x L, = x R, = Φ Γ 1,,, Γ 2 EL Γ 1,,, Γ 2 = Φ Γ 1 Γ 2 Φ Γ 1,,, 2 ER Γ 1,,, 2 = Φ 1 2 Γ 1 n 1,, n 1,, m = 1 m Table 6: From derivations to nets 16

17 = Reduce Second, we have expansions that allow to express an axiom on a compound ormula in terms o axioms on the subormulæ: Expand = = Expand Expand = = Expand = Expand Finally, (lute et al 1996) contains a large number o rewriting rules that deal with the manipulation o thinning links Fortunately, in the case o commutative logic, these rules amount to the empire rewiring proposition (Prop 33 in (lute et al 1996)), which states that the supporting wire can be chosen reely within the empire 1 o the ormula introduced by the thinning This amounts to saying that the supporting wire can be chosen reely within any net containing the original supporting wire For a detailed discussion o rewiring, see (lute et al 1996) 23 Symmetric linearly distributive categories Linearly distributive categories, which are due to Cockett and Seely and which were initially called weakly distributive categories, can be used to model MLL (This is explained in (Cockett & Seely 1997b) however, we shall spell out the semantics in 241) ll logical systems we consider in this article are commutative that is, they allow unrestricted use o the exchange rule; this allows us to use symmetric linearly distributive categories symmetric linearly distributive category (Cockett & Seely 1997b) is a 1 The empire o a ormula is the largest subnet containing that ormula as an input port or an output port 17

18 category C with two symmetric monoidal structures : C C C : C C C Ob(C) Ob(C) α : ( ) C = ( C) α : ( ) C = ( C) λ : = λ : = ρ : = ρ : = σ : σ : and a natural transormation δ : ( C) ( ) C called (linear) distribution, which must satisy various coherence conditions For a description o those conditions, see (Cockett & Seely 1997b) The distribution is used to model the cut rule, as we shall explain in 241 We call the tensor and the cotensor (Not to be conused with the cotensor product o modules) symmetric linearly distributive category with negation is a symmetric linearly distributive category together with, or every object, an object, and maps γ R : τ R : satisying the conditions below (Cockett & Seely 1997b), id τl ( ) δ ( ) γr id ρ λ id τr ( ) δ ( ) γl id ρ where γ L and τ L are the evident maps resulting rom γ R and τ R by composing with symmetry maps These maps can be used to model x L and x R, as we shall explain in 241 Symmetric linearly distributive categories with negation are equivalent to -autonomous categories (Cockett & Seely 1997b) Finally, we recall a notion that plays an important rôle in the GoI construction: a compact closed category is a symmetric linearly distributive category C with negation such that the symmetric monoidal categories (C,, ) and (C,, ) are identical, and δ is the associativity map λ 18

19 Remark 25 lternatively, one could deine a compact closed category to be a symmetric monoidal category with, or every object, an assigned let adjoint (Kelly & Laplaza 1980) The degenerate versions o the two equational laws or γ and τ are the triangular identities o that adjunction 24 Categorical semantics o MLL In this section, we recall the semantics o MLL in symmetric linearly distributive categories In 241, we describe the interpretation o derivations as morphisms In 242, we switch rom derivations to nets, because nets allows smoother presentation t the end o 242, we state the important result that MLL nets (and thereore also derivations) are in perect correspondence with symmetric linearly distributive categories (Theorem 26) 241 The interpretation o sequents n interpretation or a signature Σ in a symmetric linearly distributive category C sends every ormula over Σ to an object according to the rules = = = = I we consider the scenario with negation, then C must be a symmetric linearly distributive category with negation, and we also require = derivation Φ o a sequent 1,, n 1,, m is interpreted by a morphism Φ : 1 n 1 m, 1,, n 1,, m where and are, say, let associative, the tensor or n = 0 is, and the cotensor or m = 0 is 1 The rule x is interpreted by the identity morphism, and so are x R, x L, xr, and xl 2 The rules L resp L are interpreted by pre-composing the symmetric monoidal isomorphisms Γ ( ) Γ = Γ Γ Dually or R and R resp Γ Γ = Γ Γ 3 The rule EL is interpreted by pre-composing the symmetric-monoidal isomorphism Dually or ER Γ 1 Γ 2 = Γ 1 Γ 2 19

20 4 The cut rule is interpreted as ollows: i the interpretations o the premises are Φ = : Γ Γ 2 1,, 3 Ψ = g : Γ 1 Γ 3 2, Γ 1,, Γ 3 2 then the interpretation Φ Γ 2 1,, 3 Ψ Γ 1,, Γ 3 2 Cut o the conclusion is Γ 1, Γ 2, Γ 3 1, 2, 3 Γ 1 Γ 2 Γ 3 idid Γ1 ( 1 3 ) Γ 3 δ 1 1 ( Γ 1 Γ 3 ) 3 id g id 1 2 3, where δ 1 is obtained by combining the distribution δ and structural isomorphisms o the symmetric monoidal category (There are dierent such combinations, but the coherence laws o a symmetric linearly distributive category ensure that they all amount to the same morphism) 5 I we consider the scenario with negation, x L (resp x R) are interpreted by γ R (resp τ L ) We shall describe the semantics o weakening and contraction later in this article Evidently, an interpretation o a derivation is determined by its action on the optional axioms 242 Nets as symmetric linearly distributive categories Our goal in this section is to explain the perect correspondence between MLL and symmetric linearly distributive categories (with negation) To build a term model, we could construct a symmetric linearly distributive category whose morphisms are equivalence classes o derivations However, the range o required equational laws would be almost unmanagable, because o countless commuting conversions and laws involving the exchange rule Nets turn out to work much better here, because they deal with commuting conversions and exchange implicitly We believe that the transition rom derivations to nets is harmless, because translating derivations into nets is almost trivial (The nets can essentially be drawn into the derivation!) The aim o this section is to describe how nets can be used to construct ree symmetric linearly distributive categories (Theorems 26 and 28, which are taken rom (lute et al 1996)) 20

21 Given a set E o equivalences on Net(Σ) (where two nets can only be equivalent i they inhabit the same sequent), we can construct a symmetric linearly distributive category Net E (Σ) as ollows: The objects are the ormulæ over Σ morphism rom to is a net Net(Σ)(, ) modulo the congruence relation generated rom E and the reductions, expansions, and empire rewiring equations described in 221 Composition is deined in the evident way by connecting wires The identity morphism on is given by the wire labelled with Given nets representing morphisms : and g : C D, the net representing g is deined as below Dually or C g = C g D D The distribution is given by ( C) C δ = C ( ) C The symmetric monoidal isomorphisms with respect to are given by ( C) C α = C σ = ( ) C ρ = ρ 1 = The remaining isomorphisms α 1, λ, and λ 1 are obvious Dually or 21

22 We have the ollowing result rom (lute et al 1996): Theorem 26 Net E (Σ) is the ree symmetric linearly distributive category generated by the signature Σ and the equations E Remark 27 This theorem implies soundness and completeness when Net E (Σ) is viewed as a theory whose judgments are equalities between nets Completeness means that a judgment M = N holds in the theory Net E (Σ) whenever it holds in every model; this is true because the theory Net E (Σ) orms a model o itsel Soundness means that every interpretation o the nets over Σ in a symmetric linearly distributive category C validates the equations in 221 This is true because the canonical unctor rom Net E (Σ) to C is well-deined (ie, it sends equivalent nets to the same morphism) The ree construction can be extended with negation We only have to replace Net(Σ) with Net (Σ), allow the equations Reduce and Expand, and deine γ R = τ R = Theorem 28 Net E(Σ) is the ree symmetric linearly distributive category with negation generated by the signature Σ and the equations E 3 Modelling weakening and contraction: Dummett categories In this section, we introduce categories that are in very close correspondence with the classical sequent calculus modulo cut-reduction We proceed by extending the scenario or MLL presented in 24 with structure or modelling weakening and contraction In 31, we shall start with symmetric linearly distributive categories and add symmetric monoids and symmetric comonoids to model weakening and contraction In particular, we shall present a remarkable result (explained to us by Hasegawa) that monoids or comonoids orce symmetric linearly distributive categories to be MIX (Theorem 311) In 32, we shall add a poset-enrichment to model cut-reduction in the presence o weakening and contraction We call the resulting categories Dummett categories We do not require a Dummett category to have negation; i it has, we call it a classical category In 33, we explore the structural properties o Dummett categories In particular, we show that every hom-set o a Dummett category is a semilattice, in terms o which the poset-enrichment can be deined (Prop 328) Moreover, we show that Dummett categories have an axiomatization in terms o unconditional equalities 331 Finally, we use this to show that the construction o 22

23 the ree symmetric linearly distributive category can be extended to Dummett categories and classical categories (Theorem 332) In 34, we study the important case o compact Dummett categories (Our extended GoI construction later in the article involves only compact Dummett categories, and relies heavily on this section) Compactness allows great simpliications o the nets and the axiomatization In particular, we shall present an axiomatization o compact Dummett categories in terms o only one equality (Prop 342) Moreover, we shall show how compact Dummett categories shed light on cut-reductions involving contraction (Prop 341) Finally, we specialize the compact setting to categories with inite biproducts, explain the resulting matrix calculus, and present a single equation that characterizes when a category with inite biproducts is a Dummett category (Prop 343) 31 Symmetric monoids and comonoids To model xcl, xcr, xwl, and xwr in a symmetric linearly distributive category, we introduce maps or every object : : : [] : Deinition 31 When we use nets, we shall use the abbreviations or [] We shall require certain conditions to ensure that, [],, and are sensibly deined: we require (,, [] ) to be a symmetric monoid that is, the associativity, neutrality, and commutativity laws below have to hold ( ) α ( ) id id assoc 23

24 id [] [] id ρ λ []neutral σ symm s is easy to see, these laws correspond to the ollowing widely-accepted equalities between sequent proos: Φ Φ Γ,,,, Γ,,,, = CR applied to the let and the middle CR applied to the middle and the right Γ,,, CR Γ,, Γ,,, CR Γ,, Φ Γ,, = WR applied in either o the two evident ways Γ,,, CR Γ,, Φ Γ,, Φ Γ,,, = Φ ER Γ,,, Γ,,, The net-versions o these laws are presented in Table 7 (Owing to symmetry, we need only one o the two neutrality laws) Moreover, we require or all objects and that the monoid on is deined pointwise in terms o the monoids on and ; that is, we require id σ id pointwise, 24

25 = CR-assoc = WR-neutral = CR-symm Table 7: Net-equalities or symmetric monoids = [] and also the nullary cases [] [] []pointwise, = λ : [] = id []trivial s can be easily checked, the two nullary laws are interderivable; in the remainder o this article, we shall stick with []trivial and mention the other law no more The laws pointwise, []pointwise, and []trivial correspond to the ollowing equalities between sequent proos: Φ Γ,,,,, two applications o R Γ,,, CR Γ,, = Φ Γ,,,,, ER Γ,,,,, two applications o CR Γ,,, R Γ,, 25

26 = CR-pointwise = WR-pointwise [] = WR-trivial Table 8: Net-equalities or the pointwise deinition o the symmetric monoids Φ Γ, Φ Γ, = two applications o WR WR Γ,, Γ,,, WR Γ,, L WR = x The net-versions o these laws are presented in Table 8 Remark 32 While we believe that the laws in Table 7 are hard to dismiss (logicians seem to use them implicitly), the laws in Table 8 are perhaps more contentious We require them because they seem highly plausible and required or numerous propositions and constructions Dually, we shall use comonoids ( :, : ) to model let contraction and weakening The laws or comonoids are called assoc, []neutral, symm, pointwise, pointwise, and trivial Their netversions are called CL-assoc, WL-neutral, CL-symm, CL-pointwise, WL-pointwise, and WL-trivial 26

27 Deinition 33 symmetric monoidal category C = (C,, ) is said to have symmetric monoids i every object has a chosen symmetric monoid (, [] ), and the laws pointwise, []pointwise, and []trivial hold Dually, a symmetric monoidal category C = (C,, ) is said to have symmetric comonoids i every object has a chosen symmetric comonoid (, ), and the laws pointwise, pointwise, and trivial hold Deinition 34 pre-dummett category is a symmetric linearly distributive category C such that 1 the symmetric monoidal category (C,, ) has symmetric monoids; 2 the symmetric monoidal category (C,, ) has symmetric comonoids Remark 35 This agrees with Hasegawa s notion o pre-dummett category, except that we do not require the hom-semigroups (deined in 33) be idempotent Example 36 Every distributive lattice D The objects are the elements o D, and there is at most one morphism, which exists i and only i The unctor is the greatest lower bound, and is the least upper bound The object is the greatest element, and is the least element The distribution exists because ( C) = ( ) ( C) ( ) C The monoids and comonoids exist because =,, =, and or all D Example 37 Every symmetric monoidal category C = (C,, I) with symmetric monoids and symmetric comonoids, i both and are deined to be, and both and are deined to be I The distribution is the associativity ( C) = ( ) C Examples o such categories include: The category Rel, whose objects are (small) sets, and whose morphism are subsets o, i is deined to be the evident unctor that sends two sets to their set-theoretic product, and I is deined to be the singleton set { } We have = {((x, x), x) : x } and [] = {(, x) : x } Dually or and We write (Rel, ) or this pre-dummett category Every category with inite biproducts, i is deined to be the binary biproduct, and I is deined to be the zero (ie, initial and terminal) object The comonoids are given by the diagonals and projections o the product structure, and dually or the monoids Examples include: The category Rel, i is deined to be evident unctor that sends two sets to their disjoint union, and I is deined to be the empty set We write (Rel, ) or this pre-dummett category; The category FDVec K o inite-dimensional vector spaces over a ield K, i is deined to be the direct sum, which sends two spaces to their set-theoretic product, and I is deined to be the one-dimensional space K We write (FDVec K, ) to distinguish it rom the compact closed category based on the tensor product The product C 1 C 2 o two pre-dummett categories is a pre-dummett category Letting C 1 be a distributive lattice and C 2 be any o the categories in Example 37 shows that there exist pre-dummett categories with non-trivial hom-sets such that and 27

28 Theorem 38 Let Σ be a signature containing xwl, xwr, xcl, and xcr Let E be a set o equations on Net(Σ), and let E be the set o equations or pre-dummett categories described in Tables 7 and 8 and their duals Then Net E E (Σ) is the ree pre-dummett category generated by Σ and E Proo This ollows rom Theorem 26, and the act that E characterizes pre- Dummett categories We inish this section with some deinitions In a pre-dummett category, the morphisms π1 and π2 are deined to be id = and id =, respectively Dually, ι 1 and ι 2 are deined to be the evident morphisms and 311 MIX y MIX rule, we mean the ollowing inerence rule: Γ Γ MIX Γ, Γ, (This is the two-sided version o the MIX rule presented in (Girard 1987), not the MIX rule presented in (Gentzen 1934)) It is obviously derivable in the classical sequent calculus, or example as ollows: Γ WR Γ, Γ, Γ, Γ WL, Γ Cut (symmetric) linearly distributive category is called a (symmetric) MIX category or said to be MIX i it satisies a certain condition (which we shall present below) that ensures that the MIX rule has a canonical semantics In this section, we show that every pre-dummett category is MIX In act, we show a stronger result stating that every symmetric linearly distributive category with a monoid on or a comonoid on is MIX (Theorem 311) The MIX property is interesting rom a proo-theoretic point o view; it is also important or the equational characterization o Dummett categories ( 33) and or our extended GoI construction ( 4) (symmetric) MIX category is a (symmetric) linearly distributive category with a morphism m : such that, or all objects and, the two evident morphisms agree (Cockett & Seely 1997a): m = m 28

29 We write mix or the canonical morphism The amily mix o morphisms is easily seen to be a natural in and In a symmetric MIX category, the MIX barbell m provides a canonical way o gluing together any two nets and g: Γ Γ Γ Γ m g = m g (The supporting wire o the thinning link within each net does not matter owing to the empire rewiring proposition mentioned in 221) So a symmetric MIX category provides a canonical semantics to the MIX rule Lemma 39 Let C be a symmetric linearly category C with a morphism : Then or all, g :, we have g = g (1) Similarly when each side o Equation 1 has n = 0 or n 2 copies o as input (ie, each side has n = 0 or n 2 occurrences o ) Proo See the ppendix Lemma 310 linearly distributive category with a morphism m : is MIX i and only i the diagram below commutes m id id m = (2) = 29

30 Proo For the right-to-let direction, suppose that Diagram 2 commutes Then m = (Reduce) m = (rewiring) m = (Diagram 2) m (steps similar to the = irst three) m The let-to-right direction, which plays no rôle in this article, ollows rom simple calculations; we leave the details to the reader The ollowing theorem was ound by Hasegawa (private communications), except that we managed to remove the requirement that the comonoid (resp monoid) be symmetric Theorem 311 Every symmetric linearly distributive category with a comonoid : : is MIX (with m = ) Dually, every symmetric linearly distributive category with a monoid is MIX (with m = [] ) : [] : Proo We show the comonoid case, with the help o Lemma 310: irst, we present a net k 1 which denotes the top-right leg o Diagram 2 (with m = ), and another net k 2 which denotes the let-bottom leg; then, we use Lemma 39 to show that k 1 and k 2 are equal The dashed boxes labelled, g, h 1, and h 2 30

31 denote subnets h 1 k 1 = = (empire rewiring) = (WL-neutral) g = id = h 2 g k 2 = = (empire rewiring) = (WL-neutral) = id = y Lemma 39 with n = 2, we have h 1 = h 2, and thereore k 1 = k 2 Corollary 312 Every pre-dummett category is MIX with m =, and also with m = [] We write mix (resp mix[] ) or the natural transormation built rom (resp [] ) re pre-dummett categories canonically MIX that is, do we have = []? We do not know the general answer to this question, but we shall see (Lemma 323) that the answer or Dummett categories is yes 32 Poset-enrichment Our next goal is to model the cut-reduction rules or weakening and contraction that is, the equations in Table 9 and their duals ReduceWR and 31

32 Φ Γ, Γ,Γ, Ψ Γ WL,Γ Cut Ψ Γ WL, WR Γ, Γ, ReduceWL Φ Γ, Ψ,, Γ CL,Γ Cut Φ Γ, Φ Γ, Ψ,, Γ Cut,Γ, Γ, Cut Γ,Γ, Γ,Γ, Γ,, CL,CR Γ,Γ, ReduceCL Table 9: Cut-reductions or weakening and contraction (representative cases) ReduceCR We use the symbol instead o the equality symbol, because we shall not require that the denotation o redex and reduct be the same: i we required them to be the same in the rules ReduceWL and ReduceWR, then any two derivations o Γ would have the same denotation because o Laont s example; i we required them to be the same in the rules ReduceCL and ReduceCR, we would rule out desirable models, as we shall see in Example 315 Table 10 contains the net-versions o the reductions in Table 9 The derivation Φ corresponds to the net The net corresponding to Ψ is not needed, because we allow ourselves to rewrite subcircuits We assume without loss o generality that Γ and consist o single ormulæ; this is possible because we can always bundle a wire labelled with Γ = 1,, n in a single wire labelled with 1 n (by using the two kinds o links or ), and a wire labelled with = 1,, m in a single wire labelled with 1 m (by using the two kinds o links or ) In our categorical models, will be a poset-enrichment Consider the netversion o the law ReduceCL; i is empty, it corresponds to categorical law ( ) Γ lax, which states that is a lax natural transormation Similarly, the law ReduceWL or empty corresponds to the categorical law Γ lax, which states that be a lax natural transormation The parametric categorical laws that is, the versions or non-empty, are very cumbersome: stating them requires multiple uses o the distribution δ; alternatively, one can stick with the non-parametric versions and add our extra inequalities (see Table 1 in (Führmann & Pym 2004a)) y contrast, the netversions o the laws are elegant; moreover, equations between nets are perectly 32

33 Γ Γ ReduceWL Γ Γ ReduceCL Table 10: Net-version o Table 9 suitable to describe equalities between morphisms, owing to Theorem 26 So we stick with the net-versions in this article Deinition 313 Dummett category is a pre-dummett category C together with a poset-enrichment such that 1 the unctors and are monotonic in both arguments; 2 the laws ReduceWL, ReduceWR, ReduceCL, and ReduceCR hold Example 314 Every distributive lattice D (which we know to be a pre- Dummett category rom Example 36) The partial order is trivial, because each hom-set contain at most one element Example 315 The pre-dummett category (Rel, ), where or relations, :, we deine, where is the set-theoretic inclusion To see that ReduceWL holds, let Γ,,, be sets, and let be a relation Γ The relation denoted by the redex turns out to be {((g, b), (d, b)) : b a : (g, (d, a)) }, while the reduct turns out to be {((g, b), (d, b)) : b g Γ d } The two are equal i and only i or all g Γ and d, there exists an a such that (g, (d, a)) We call such relations : Γ total; or empty, this agrees with the usual notion o total relation Dually or ReduceWR To see that ReduceCL holds, note that reduct turns out to be {(g, (d, a, a)) : (g, (d, a)) }, 33